Encyclopedia/All topics/Foundation

Foundation

Articles 2,281–2,340 of 2,979. Alphabetical by title.

• Foundation Primitive Recognition Calculus Real Complete Ordered Field Prcreal Ne

  A machine-checked proof that negating two equivalent sequences of rational numbers keeps them equivalent, a step toward building the real numbers from recognition events.

• Foundation Primitive Recognition Calculus Real Complete Ordered Field Promoted

  A machine-checked certificate confirms that the framework's internal real numbers already carry the operations and theorems of a complete ordered field.

• Foundation Primitive Recognition Calculus Real Complete Ordered Field Promoted P

  A machine-checked certificate confirms that a primitive internal number system already carries the structure of the real numbers, with full typeclass instances deferred to a later

• Foundation Primitive Recognition Calculus Real Completeness

  Real numbers in Recognition Science are built from rational sequences, and completeness means every such sequence has a limit within the same construction.

• Foundation Primitive Recognition Calculus Real Completeness Prc Real Completenes

  A machine-checked proof shows that within a primitive calculus of rational records, every Cauchy sequence has a limit, a completeness property that makes the system behave like the

• Foundation Primitive Recognition Calculus Real Completeness Prcjcost Distance Th

  A machine-checked theorem shows that a specific way of measuring distance between rational numbers is continuous, a key step in building real numbers from a ledger of recognition e

• Foundation Primitive Recognition Calculus Real Completeness Prcreal Completeness

  A machine-checked theorem shows that picking a diagonal from a grid of approximations is enough to guarantee that every Cauchy sequence of rationals converges to a real number.

• Foundation Primitive Recognition Calculus Real Completeness Prcreal Diagonal Sel

  A machine-checked proof shows that a sequence of rational approximations always has a point of the real line as its limit, by picking one entry from each row of an infinite table.

• Foundation Primitive Recognition Calculus Real Completeness Prcreal Finite Repre

  A machine-checked proof shows that any converging sequence of rationals has a limit that can be found by reading only finitely many terms at each stage.

• Foundation Primitive Recognition Calculus Real Completeness Prcreal Raw Diagonal

  A machine-checked proof shows that any orderly list of rational sequences has a single diagonal sequence that captures its limit, a step toward building real numbers from recogniti

• Foundation Primitive Recognition Calculus Real Completion

  A discrete counting system reaches the continuous real number line, and the move is honestly labeled as a choice, not a forced step.

• Foundation Primitive Recognition Calculus Real Completion Complete Space

  The framework's first complete space is not a new construction: it is the ordinary real number line, imported and tagged as a classical extension.

• Foundation Primitive Recognition Calculus Real Completion Real Completion Bounda

  The real numbers enter Recognition Science as a classical extension, not an internal construction, and the certificate says so plainly.

• Foundation Primitive Recognition Calculus Real Completion Real Completion Claim

  A machine-checked certificate records the first step from rational recognition tokens to the complete real line, and it is honest about what remains classical.

• Foundation Primitive Recognition Calculus Real Line Non Nativity Faithful Cover

  A machine-checked theorem draws a sharp line: a set can be faithfully labeled by whole numbers exactly when it is countable, and the real line falls on the far side.

• Foundation Primitive Recognition Calculus Real Line Non Nativity No Faithful Cov

  A machine-checked proof shows that no countable system of distinct labels can tag every real number, a cardinality wall that separates what recognition can witness from what it can

• Foundation Primitive Recognition Calculus Real Line Non Nativity Real Not Faithf

  The real number line cannot be fully labeled by any countable system of distinct certificates; this is a proved cardinality fact, not a claim about physics.

• Foundation Primitive Recognition Calculus Real Mul Bounded Continuity

  A machine-checked library proves that multiplying real numbers on the recognition ledger works, provided the ledger entries eventually stay within a finite bound.

• Foundation Primitive Recognition Calculus Real Mul Bounded Continuity Prc Real M

  A machine-checked certificate shows that multiplying real numbers in one framework's calculus reduces to two simpler conditions, but it does not prove those conditions hold.

• Foundation Primitive Recognition Calculus Real Mul Bounded Continuity Prccauchy

  A Cauchy sequence of rational numbers is eventually bounded; the Recognition Science library states this as a target for its primitive ledger sequences.

• Foundation Primitive Recognition Calculus Real Mul Bounded Continuity Prcrat

  Multiplying infinite sequences in a framework where recognition costs are forced needs a guarantee that the product stays finite; this page explains that guarantee.

• Foundation Primitive Recognition Calculus Real Mul Bounded Continuity Prcraw Eve

  A Cauchy sequence of rational numbers eventually stays inside a finite interval; this property is what lets multiplication of real numbers be defined consistently.

• Foundation Primitive Recognition Calculus Real Mul Bounded Continuity Prcreal Mu

  A single theorem in the framework's library shows that multiplying real numbers stays consistent as long as two modest analytic conditions hold, and it names those conditions

• Foundation Primitive Recognition Calculus Real Null Setoid

  How a formal calculus builds real numbers from a recognition cost, and the one analytic step still needed to finish the construction.

• Foundation Primitive Recognition Calculus Real Null Setoid Prcjcost Distance Tri

  A formal target that, if proved, would let the framework treat points at zero distance as equivalent, and why that matters for building real numbers.

• Foundation Primitive Recognition Calculus Real Null Setoid Prcnull Distance Seto

  A machine-checked theorem shows one local analytic condition is enough to build a real number system from a discrete recognition ledger.

• Foundation Primitive Recognition Calculus Real Null Setoid Prcnull Distance Tran

  A single analytic condition turns a formal notion of "zero distance" into a proper equivalence relation, the last step before building a real-number-like structure.

• Foundation Primitive Recognition Calculus Real Null Setoid Prcreal Null Setoid C

  A machine-checked certificate that says: once one analytic inequality is proved, the rest of the real-number construction follows automatically.

• Foundation Primitive Recognition Calculus Real Null Setoid Real Null Setoid Clai

  A formal certificate records exactly which unproved step still blocks the construction of real numbers from recognition events.

• Foundation Primitive Recognition Calculus Real Null Setoid Real Null Setoid Cond

  A machine-checked certificate shows that building real numbers from recognition cost needs exactly one more analytic proof, and no new machinery.

• Foundation Primitive Recognition Calculus Real Order Congruence

  Real order congruence is a consistency condition: it guarantees that the ordering of real numbers does not depend on which approximating sequence you use to represent them.

• Foundation Primitive Recognition Calculus Real Order Congruence Prc Real Order C

  In the Recognition Science framework, a machine-checked certificate proves that the real-order comparison of recognition costs is well-defined: equivalent sequences compare the sam

• Foundation Primitive Recognition Calculus Real Order Congruence Prcjcost Distanc

  A small measured cost between two recognized events forces their underlying values to be close, with a precise bound that the framework proves.

• Foundation Primitive Recognition Calculus Real Order Congruence Prcraw Eventuall

  In building real numbers from recognition sequences, this theorem shows that 'eventually no larger' is a well-defined comparison, not an artifact of how a sequence is rep

• Foundation Primitive Recognition Calculus Real Order Congruence Prcreal Order Co

  When two descriptions of the same real number are interchangeable, the order between numbers must stay the same; this theorem proves that it does.

• Foundation Primitive Recognition Calculus Real Order Congruence Rat Sq Lt Sq Bou

  A small lemma about squares of rational numbers turns out to be the load-bearing step that lets a discrete recognition ledger inherit the usual ordering of real numbers.

• Foundation Primitive Recognition Calculus Real Product Continuity

  A machine-checked proof that the cost function's basic arithmetic operation, multiplying two real-valued recognition states, behaves continuously.

• Foundation Primitive Recognition Calculus Real Product Continuity Prc Real Produ

  A machine-checked proof that multiplication of real numbers stays continuous when the numbers are built from a discrete recognition ledger.

• Foundation Primitive Recognition Calculus Real Product Continuity Prcjcost Dista

  A machine-checked proof establishes that a specific distance function in the framework's ledger is continuous under multiplication, a key step toward building real numbers fro

• Foundation Primitive Recognition Calculus Recognizer Bridge

  A bridge that connects the primitive recognition calculus to the existing uniqueness theorem, closing the loop on how recognition costs are forced.

• Foundation Primitive Recognition Calculus Recognizer Bridge Cost To Rat

  A single formula converts any positive ratio into a recognition cost, and the formula is proved, not assumed.

• Foundation Primitive Recognition Calculus Recognizer Bridge Cost To Real Jcost

  A small theorem in a machine-checked library connects the discrete recognition ledger to the real-number cost function, but the full story remains open.

• Foundation Primitive Recognition Calculus Recognizer Bridge Prc Recognizer Bridg

  A machine-checked certificate shows that a primitive recognition calculus connects to a proved uniqueness theorem, while leaving a fully native proof as an open target.

• Foundation Primitive Recognition Calculus Recognizer Bridge Prcpositive Ratio

  A positive ratio is the basic input to a recognition cost, and the framework proves the cost formula that any such input must obey.

• Foundation Primitive Recognition Calculus Recognizer Bridge Prcrecognizer Law Of

  A machine-checked proof shows that any cost function satisfying five plain conditions must equal one specific formula, and it says nothing about costs that fail those conditions.

• Foundation Primitive Recognition Calculus Rigidity Base Initiality Base Categori

  A theorem in the Recognition Science library proves that the natural numbers are the only structure with a starting point and a distinct next step, up to a unique relabeling.

• Foundation Primitive Recognition Calculus Rigidity Base Initiality Base Initial

  A minimal counting structure, with only a starting point and a next step, turns out to be unique: any two such structures are the same.

• Foundation Primitive Recognition Calculus Rigidity Base Initiality Base Rec Inje

  A machine-checked proof shows that any system of discrete steps that obeys the basic rules of counting must be the same, in a precise sense, as the natural numbers.

• Foundation Primitive Recognition Calculus Rigidity Base Initiality Base Rec Surj

  A simple counting argument, machine-checked, shows that any structure obeying the natural-number laws must be exactly the counting numbers, no more and no less.

• Foundation Primitive Recognition Calculus Rigidity Base Initiality Base Rigidity

  A machine-checked theorem shows that any structure obeying the simple rules of counting is forced to be the natural numbers, and nothing else.

• Foundation Primitive Recognition Calculus Rigidity Base Initiality Hom Eq Base R

  A theorem about counting shows that any system that behaves like the natural numbers is forced to be identical to them, a rigidity result with a simple proof.

• Foundation Primitive Recognition Calculus Rigidity Base Initiality Is Peano Mode

  A small set of axioms pins down the natural numbers as the unique structure for counting distinctions, and the framework's library proves it.

• Foundation Primitive Recognition Calculus Rigidity Ledger Transport

  A machine-checked library proves that truths about the natural numbers stay true in any model of counting, with no extra assumptions.

• Foundation Primitive Recognition Calculus Rigidity Ledger Transport Add Comm Tra

  A machine-checked proof that adding distinctions commutes, once verified in one model, carries over to every structure that behaves like the counting numbers.

• Foundation Primitive Recognition Calculus Rigidity Ledger Transport Base To Nat

  A machine-checked proof shows that the framework's primitive ledger of distinctions is exactly the natural numbers, with no extra assumptions.

• Foundation Primitive Recognition Calculus Rigidity Ledger Transport Msat All Cov

  A machine-checked proof shows that a formula true in the framework's canonical model remains true in any other model that satisfies the same structural conditions, with no ext

• Foundation Primitive Recognition Calculus Rigidity Ledger Transport Nat Algebra

  In Recognition Science, the natural numbers are not an encoding choice but the unique model of a primitive distinction calculus, a fact its machine-checked library proves.

• Foundation Primitive Recognition Calculus Rigidity Ledger Transport Nat Rec Inje

  A single injective map is the load-bearing wall that lets one model of counting stand in for another.

• Foundation Primitive Recognition Calculus Rigidity Ledger Transport Nat Rec Surj

  A machine-checked proof shows that counting steps in any structure that behaves like the natural numbers reaches every element, with no hidden choices.

• Foundation Primitive Recognition Calculus Rigidity Ledger Transport Transport Gr

  A proof's validity can move between mathematical universes without carrying any extra assumptions, a result with a precise limit.